Quantum manifestations of classical chaos in a Fermi accelerating disk

We study the classical and quantum mechanics of a two-dimensional version of a Fermi accelerator. The model consists of a free particle that collides elastically with the walls of a circular disk with the radius varying periodically in time. A complete quantum mechanical solution of the problem is possible for a specific choice of the time-periodic oscillating radius. The quasi-energy spectral properties of the model are obtained from direct evaluation of finitedimensional approximations to the time evolution operator. As the scaled h̄ is changed from large to small the statistics of the Quasienergy Eigenvalues (QEE) change from Poisson to circular orthogonal ensemble (COE). Different statistical tests are used to characterize this transition. The transition of the Quasienergy Eigenfunctions (QEF ) is also studied using the χ2 test with ν degrees of freedom. The Porter-Thomas distribution is shown to apply in the COE regime, while the Poisson regime does not fit the χ2 test with ν = 0. We find that the Poisson regime is associated with exponentially localized QEF whereas the eigenfunctions are extended in the COE regime. 1 To make a direct comparison between the classical and quantum solutions we change the representation of the model to one in which the boundary is fixed and the Hamiltonian acquires a quadratic term with a time-periodic frequency. We then carry out a successful comparison between specific classical phase space surface-of-section solutions and their corresponding quasi-energy eigenfunctions in the Husimi representation.